The 3x+1 semigroup Original Research Article

The 3x+1 semigroup is the multiplicative semigroup image of positive rational numbers generated by image together with {2}. This semigroup encodes backwards iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals image in lowest terms with image, and so contains all positive integers.